- Asked 2/25
- 2075 Chaitra · 8 marks
Bezier points at u=0.25,0.5,0.75: (10,10)...
Short answer
With control points (10,10),(15,25),(20,30),(25,5), the cubic Bezier gives P(0.25)=(13.75,19.06), P(0.5)=(17.5,22.5), P(0.75)=(21.25,18.44). The curve starts at (10,10), rises through these points, and ends at (25,5), inside the convex hull.
Find the coordinates at U=0.25, 0.5, and 0.75 with respect to the control points (10,10), (15,25), (20,30), and (25,5) using Bezier function. Draw your curve with given control points.
Control points P0(10,10), P1(15,25), P2(20,30), P3(25,5). Cubic Bezier:
P(u) = (1−u)³P0 + 3u(1−u)²P1 + 3u²(1−u)P2 + u³P3
| u | B0 | B1 | B2 | B3 |
|---|---|---|---|---|
| 0.25 | 0.4219 | 0.4219 | 0.1406 | 0.0156 |
| 0.50 | 0.1250 | 0.3750 | 0.3750 | 0.1250 |
| 0.75 | 0.0156 | 0.1406 | 0.4219 | 0.4219 |
u = 0.25 x = 0.4219·10 + 0.4219·15 + 0.1406·20 + 0.0156·25 = 4.219 + 6.328 + 2.813 + 0.391 = 13.75 y = 0.4219·10 + 0.4219·25 + 0.1406·30 + 0.0156·5 = 4.219 + 10.547 + 4.219 + 0.078 = 19.06
u = 0.5 x = 0.125·10 + 0.375·15 + 0.375·20 + 0.125·25 = 1.25 + 5.625 + 7.5 + 3.125 = 17.5 y = 0.125·10 + 0.375·25 + 0.375·30 + 0.125·5 = 1.25 + 9.375 + 11.25 + 0.625 = 22.5
u = 0.75 x = 0.0156·10 + 0.1406·15 + 0.4219·20 + 0.4219·25 = 0.156 + 2.109 + 8.438 + 10.547 = 21.25 y = 0.0156·10 + 0.1406·25 + 0.4219·30 + 0.4219·5 = 0.156 + 3.516 + 12.656 + 2.109 = 18.44
Answer
| u | x | y |
|---|---|---|
| 0 | 10 | 10 |
| 0.25 | 13.75 | 19.06 |
| 0.50 | 17.50 | 22.50 |
| 0.75 | 21.25 | 18.44 |
| 1 | 25 | 5 |
Drawing: plot P0 to P3 and join them with straight lines (the control polygon). Then plot the points in the table and join them smoothly. The curve starts at (10,10) heading towards P1, rises to about y = 22.5 in the middle, and falls to (25,5), staying inside the convex hull of the four control points.
30 | o P2
|
25 | o P1
| *
20 | *
| *
15 |
|
10 |o P0
|
5 | o P3
+-------------------------------- x
10 15 20 25
o = control points, * = calculated points at u = 0.25, 0.5, 0.75
✓ Answer checked by Claude Opus 5.5 (high effort)
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